"Fractal Geometry is not just a chapter of mathematics, but one that helps Everyman to see the same old world differently". - Benoit Mandelbrot

The world of mathematics usually tends to be thought of as abstract. Complex and imaginary numbers, real numbers, logarithms, functions, some tangible and others imperceivable. But these abstract numbers, simply symbols that conjure an image, a quantity, in our mind, and complex equations, take on a new meaning with fractals - a concrete one. Fractals go from being very simple equations on a piece of paper to colorful, extraordinary images, and most of all, offer an explanation to things. The importance of fractal geometry is that it provides an answer, a comprehension, to nature, the world, and the universe. Fractals occur in swirls of scum on the surface of moving water, the jagged edges of mountains, ferns, tree trunks, and canyons. They can be used to model the growth of cities, detail medical procedures and parts of the human body, create amazing computer graphics, and compress digital images. Fractals are about us, and our existence, and they are present in every mathematical law that governs the universe. Thus, fractal geometry can be applied to a diverse palette of subjects in life, and science - the physical, the abstract, and the natural.

We were all astounded by the sudden revelation that the output of a very simple, two-line generating formula does not have to be a dry and cold abstraction. When the output was what is now called a fractal, no one called it artificial... Fractals suddenly broadened the realm in which understanding can be based on a plain physical basis. (McGuire, Foreword by Benoit Mandelbrot)

A fractal is a geometric shape that is complex and detailed at every level of magnification, as well as self-similar. Self-similarity is something looking the same over all ranges of scale, meaning a small portion of a fractal can be viewed as a microcosm of the larger fractal. One of the simplest examples of a fractal is the snowflake. It is constructed by taking an equilateral triangle, and after many iterations of adding smaller triangles to increasingly smaller sizes, resulting in a "snowflake" pattern, sometimes called the von Koch snowflake. The theoretical result of multiple iterations is the creation of a finite area with an infinite perimeter, meaning the dimension is incomprehensible. Fractals, before that word was coined, were simply considered above mathematical understanding, until experiments were done in the 1970's by Benoit Mandelbrot, the "father of fractal geometry". Mandelbrot developed a method that treated fractals as a part of standard Euclidean geometry, with the dimension of a fractal being an exponent.

Fractals pack an infinity into "a grain of sand". This infinity appears when one tries to measure them. The resolution lies in regarding them as falling between dimensions. The dimension of a fractal in general is not a whole number, not an integer. So a fractal curve, a one-dimensional object in a plane which has two-dimensions, has a fractal dimension that lies between 1 and 2. Likewise, a fractal surface has a dimension between 2 and 3. The value depends on how the fractal is constructed. The closer the dimension of a fractal is to its possible upper limit which is the dimension of the space in which it is embedded, the rougher, the more filling of that space it is. (McGuire, p. 14)

Fractal Dimensions are an attempt to measure, or define the pattern, in fractals. A zero-dimensional universe is one point. A one-dimensional universe is a single line, extending infinitely. A two-dimensional universe is a plane, a flat surface extending in all directions, and a three-dimensional universe, such as ours,...

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...﻿FRACTALGEOMETRY
INTRODUCTION
Fractals is a new branch of mathematics and art. Most physical systems of nature and many human artifacts are not regular geometric shapes of the standard geometry derived from Euclid (i.e, Euclidian geometry: comprising of lines, planes, rectangular volumes, arcs, cylinders, spheres, etc.) Fractalgeometry offers almost unlimited ways of describing, measuring and predicting the natural phenomena.
“Why is geometry often described as ‘cold and dry’? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
(Benoit Mandelbrot)
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...The Application of FractalGeometry to Ecology
Principles of Ecology 310L
Victoria Levin
7 December 1995
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New insights into the natural world are just a few of the results from the use
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...FractalGeometry
How would you like to take a class called geometry of chaos? Probably doesn’t sound too thrilling. A man named Benoit Mandelbrot is responsible for creating the geometry of chaos. The geometry of chaos is considered to be the fourth-dimension. It is considered to be the world in which we live in, a world where there is constant change based on feedback, an open system where everything is related to everything else. It is now recognized as the true geometry of nature. The geometric system the can describe the simple shapes of the world (Lauwerier).
Fractalgeometry is a structure that provided a new key for the study of non-linear processes (Lauwerier). Benoit Mandelbrot explained that lines have a single dimension, plane figures have two dimensions and that we live in a three dimensional spatial world (Fractals Useful Beauty). In a paper published in 1967, Mandelbrot investigated the idea of measuring the length of a coastline. Mandelbrot explained that the shape of a coastline defies conventional Euclidean geometry and that rather than having a natural number dimension, it has a “fractional dimension.” The coastline is an example of a self-similar shape, which is a shape that repeats itself over and over on different scales (Fractals).
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...Geometry in everyday life
Geometry was thoroughly organized in about 300bc, when the Greek mathematician, Euclid gathered what was known at the time; added original work of his own and arranged 465 propositions into 13 books, called Elements.
Geometry was recognized to be not just for mathematicians. Anyone can benefit from the basic learning of geometry, which is to follow the lines reasoning. Geometry is one of the oldest sciences and is concerned with questions of shape, size and relative position of figures and with properties of space.
Geometry is considered an important field of study because of its applications in daily life.
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...﻿Fractals
Introduction
Fractals are geometric patterns that when repeated at increasingly smaller scales they produce irregular shapes and surfaces. All fractals have a feature of ‘self-similarity’. A set is self-similar if it can be broken into arbitrary small pieces, each of which is a small copy of the entire set, for fractals the pattern reproduced must be detailed (Nuhfer 2006). Self-similarity may be demonstrated as exact self-similarity meaning the fractal is identical at all scales a fractal that demonstrates exact self-similarity is the Koch Snowflake. Other fractals exhibit quasi self-similarity. This is when fractals approximate the same pattern at different scales, they contain small copies of the entire fractal in altered or degenerate forms, and an example of this is the Mandelbrot set (Fractal 2009). Also, fractal curves are ‘nowhere differentiable’ meaning that the gradient of the curve can never be found; because of this fractals cannot be measured in traditional ways (Turner 1998). I find it interesting to note that many phenomena in nature have fractal features including clouds, mountains, fault lines and coastlines. There are also a range of mathematical structures that are fractals including, Sierpinski triangle, Koch snowflake, Peano curve and the...

...Fractals have been one of the tools used in Euclidean geometry to explain the abnormal shapes in nature. Fractals are able to explain the irregular shapes that are a far cry from the normal circle or square. It is an object of symmetry that uses components to create the picture of a self-similar entity.
Fractals first appeared on the scene in 1918 due to the mathematician, Felix Hausdroff. A Poland mathematician by the name of Beniot B. Mandelbrot began the term fractals. Fractals originated from the Latin term fractus meaning broken or fractured. It is a series of self-similar images repeated; The Koch snowflake, the Mandelbrot set, the Julia set and the Box fractal are many examples.
The idea of a fractal is a pattern of repetitive images of the entire picture. When magnified upon, the image continues to look the same and builds upon the whole picture. “A key characteristic of fractals is fractal dimension.” [http://www.reference.com/browse/wiki/Fractal] This is the parameter of the fractal that uses fractions or nonintergers. “The table below shows the complexity of a figure as it increases its dimension.” [http://library.thinkquest.org/26242/full/index.html.]
F A finite number greater than 0
I An infinite number
Dimension Num of Points Length Area Volume
D = 0 F 0 0 0
0 < D < 1 I 0 0 0
D = 1 I...

...1. Introduction
The birth of every technology is the result of the quest for automation of some form of human work. This has led to many inventions that have made life easier for us. Fractal Robot is a science that promises to revolutionize technology in a way that has never been witnessed before.
The principle behind Fractal Robots is very simple. You take some cubic bricks made of metals and plastics, motorize them, put some electronics inside them and control them with a computer and you get machines that can change shape from one object to another. Almost immediately, you can now build a home in a matter of minutes if you had enough bricks and instruct the bricks to shuffle around and make a house! It is exactly like kids playing with Lego bricks and making a toy hose or a toy bridge by snapping together Lego bricks-except now we are using computer and all the work is done under total computer control. No manual intervention is required. Fractal Robots are the hardware equivalent of computer software.
1. What are Fractals?
A fractal is anything which has a substantial measure of exact or statistical self-similarity. Wherever you look at any part of its body it will be similar to the whole object.
2. Fractal Robots
A Fractal Robot physically resembles itself according to the definition above. The robot can be animated around its joints in a...